
Proof: Proposition 6 in [1]} ([1]}) proves that \(\rho ^{\lceil M \rceil }_{i,m}\)  converges to \(\rho _{i,m}\)  in probability for all \(m \in \mathbb {N}\) . Proposition 7 in [1]} ([1]}) proves that \( \vert \rho ^{\lceil M \rceil }_{i,m}-\widehat{\rho }_{i,m} \vert =O_p(M^{3/2}_{\rm max}\max (n^{-1/2},\triangle _M r_n))\) , where \(M_{\rm max}=\max \limits _{j=1,\ldots ,p}M_j\) , \(r_n= n^{−1}h^{-2}\)  in the case of measurement error or irregularly sampled data for a certain bandwidth \(h\) . Therefore \(\widehat{\rho }_{i,m}\)  converges to \(\rho _{i,m}\)  in distribution. We can regard \(\widehat{\rho }_{i,m}\)  as a conditional expectation of \(\rho _{i,m}\)  given \(\mathbf {Y}_i\) . Assume that \(\rho _{i,m}\)  and \(\mathbf {\epsilon }_{i}\)  are jointly Gaussian, \(\rho _{i,m}\)  and \(\mathbf {Y}_{i}\)  are also jointly Gaussian, so do \(\mathbf {\rho }_{iM}\)  and \(\mathbf {Y}_{i}\) . var\((\widehat{\mathbf {\rho }}_{iM})={\rm cov}({\mathbf {\rho }}_{iM},{\mathbf {Y}}_i)\mathbf {\Sigma }^{-1}_{{\mathbf {Y}}_i}{\rm cov}({\mathbf {\rho }}_{iM},{\mathbf {Y}}_i)^\top =\mathbf {V}\mathbf {\Psi }_{iM}^\top \mathbf {\Sigma }^{-1}_{{\mathbf {Y}}_i}\mathbf {\Psi }_{iM}\mathbf {V}\) , \({\mathbf {\Sigma }}_{{\mathbf {Y}}_i}=\mathbf {\Psi }_{iM}\mathbf {V\mathbf {\Psi }^\top _{iM}}+\hbox{diag}\lbrace \sigma _1^2 \mathbf {I}_{N_{1i}},\ldots , \sigma _p^2\mathbf {I}_{N_{pi}}\rbrace \) . \({\rm cov}(\widehat{\mathbf {\rho }}_{iM},\mathbf {\rho }_{iM})={\rm var}(\widehat{\mathbf {\rho }}_{iM})\)  because \(\widehat{\mathbf {\rho }}_{iM}\)  is the projection of \(\mathbf {\rho }_{iM}\)  on the space spanned by the linear functions of \({\mathbf {Y}}_i\) . Then var\((\widehat{\mathbf {\rho }}_{iM}-\mathbf {\rho }_{iM})={\rm var}(\widehat{\mathbf {\rho }}_{iM})+{\rm var}(\mathbf {\rho }_{iM})-2{\rm cov}(\widehat{\mathbf {\rho }}_{iM},\mathbf {\rho }_{iM})={\rm var}(\mathbf {\rho }_{iM})-{\rm var}(\widehat{\mathbf {\rho }}_{iM})=\mathbf {\Omega }_{M}\) . Under Gaussian assumptions, \((\widehat{\mathbf {\rho }}_{iM}-\mathbf {\rho }_{iM})\sim \mathcal {N}_M(0,\mathbf {\Omega }_{M})\)  when \(n \rightarrow \infty \) . \(\Box \)
